The paper under review is the first part of authors’ recent investigation on the correlation of the von Mongoldt and higher divisor functions. Both papers concerned with the asymptotic estimation of correlation of the form \(\sum_{X<n\leq 2X}f(n)\overline{g(n+h)}\) where \(f\) and \(g\) taken as von Mongoldt function \(\Lambda\) and higher divisor function \(d_k\) defined by \(d_k(n)=\sum_{n_1\cdots n_k=n}1\), for which estimation of correlations are related to Hardy-Littlewood prime tuples conjecture, divisor correlation conjecture, higher order Titchmarsh divisor problem, and quantitative Goldbach conjecture. More precisely, let \(\sigma=\frac{8}{33}\), \(A>0\), \(0<\varepsilon<\frac{1}{2}\) and \(k,l\geq 2\) be fixed, and suppose that \(X^{\sigma+\varepsilon}\leq H\leq X^{1-\varepsilon}\) for some \(X\geq 2\). Also, let \(0\leq h_0\leq X^{1-\varepsilon}\). In this paper, the authors prove the following results:

(i)

(Related by Hardy-Littlewood prime tuples conjecture) One has \[ \sum_{X<n\leq 2X}\Lambda(n)\Lambda(n+h)=\mathfrak{S}(h)X+O_{A,\varepsilon}(X\log^{-A}X), \] for all but \(O_{A,\varepsilon}(H\log^{-A}X)\) values of \(h\) with \(|h-h_0|\leq H\). The singular series \(\mathfrak{S}(h)\) vanishes if \(h\) is odd, and is to \[ \mathfrak{S}(h)=2\prod_p\left(1-\frac{1}{(p-1)^2}\right)\prod_{p|h:p>2}\frac{p-1}{p-2}, \] when \(h\) is even.

(ii)

(Related by divisor correlation conjecture) For some polynomial \(P_{k,l,h}\) of degree \(k+l-2\) one has \[ \sum_{X<n\leq 2X}d_k(n)d_l(n+h)=P_{k,l,h}(\log X)X+O_{A,\varepsilon,k,l}(X\log^{-A}X), \] for all but \(O_{A,\varepsilon,k,l}(H\log^{-A}X)\) values of \(h\) with \(|h-h_0|\leq H\).

(iii)

(Related by higher order Titchmarsh divisor problem) For some polynomial \(Q_{k,h}\) of degree \(k-1\) one has \[ \sum_{X<n\leq 2X}\Lambda(n)d_k(n+h)=Q_{k,h}(\log X)X+O_{A,\varepsilon,k}(X\log^{-A}X), \] for all but \(O_{A,\varepsilon,k}(H\log^{-A}X)\) values of \(h\) with \(|h-h_0|\leq H\).

(iv)

(Related by Goldbach conjecture) One has \[ \sum_{n}\Lambda(n)\Lambda(N-n)=\mathfrak{S}(N)N+O_{A,\varepsilon}(X\log^{-A}X), \] for all but \(O_{A,\varepsilon}(H\log^{-A}X)\) integers \(N\) in the interval \([X,X+H]\).

The authors end the paper by an appendix providing the proof of a key proposition by using combinatorial decomposition, mean-value theorems and large value theorems for Dirichlet polynomials.